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Continuous time QMC: application for DMFT and beyond A.N. Rubtsov Moscow Sta

Continuous time QMC: application for DMFT and beyond A.N. Rubtsov Moscow State University. An impurity problem of DMFT. Discrete time in Hirsh-Fye QMC. Hubbard-Stratonovich transformation. The idea. Gaussian.

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Continuous time QMC: application for DMFT and beyond A.N. Rubtsov Moscow Sta

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  1. Continuous time QMC: application for DMFT and beyondA.N. Rubtsov Moscow State University

  2. An impurity problem of DMFT

  3. Discrete time in Hirsh-Fye QMC Hubbard-Stratonovich transformation The idea Gaussian

  4. An idea behind continuous-time QMC Series expansion in powers of some perturbation W Let’s do a random walk in the space of all i, j, t and perturbation order • The series must converge • Integrand should be “at most positive”

  5. Flavours of continuous-time QMC Path-integral method Finite perturbation order is visible only for discrete lattice Terrible sign problem for fermions

  6. Weak-coupling Series Expansion Divide action into Gaussian part and perturbation example: single Hubbard atom with attraction Treat W as a perturbation. In the interaction representation For fermions, perturbation series converge. Idea of algorithm is to perform a random walk in the space of {k, (arguments of integrals)} .

  7. Random walk in K-space Call state K the set and perform a random walk with probability density Trace here can be explicitly calculated from the Wick theorem(H0 is Gaussian!) It is possible to prove that for such an ensemble

  8. Strong-coupling expansion P. Werner et. al, PRL 98, 070602 (2007), PRB 75, 085108 (2007) Exact expression for Hamiltonian system

  9. Calculation with different CT-QMC schemes E. Gull et al cond-mat/0609438

  10. Back to weak-coupling expansion:Optimization of the average sign Recall the formula for isolated Hubbard atom Positive U results in the alternative signs (unacceptable sign problem)! For half-filling, particle-hall transformation eliminates the “trivial” sign problem: The split into Gaussian part and perturbation is not unique. The average sign can be optimized by the proper choice of this split. Typically we recommend the perturbation of the type U (n-a) (n-a), 0<a<1 for U<0 U (n+a) (n-1-a) or 0.5 U (n+a) (n-1-a)+ 0.5 U (n-1-a) (n+a) a<<1 for U<0 We observed that slight variations of a within mentioned intervals does not affect the average sign remarkably But particular choice of a is still a matter of art!

  11. Worked example: 2x2 Hubbard cluster 2x2 Hubbard lattice away from half-filling: three electrons in the system. Average sign as a function of b: CT-QMC (filled symbols) and auxiliary-field quantum Monte Carlo (opened symbols) algorithms. Lines are guides to the eye. Parameters: U = 4, t = 1.

  12. Metal-insulator transition in the Hubbard model on Bethe lattice DMFT on Bethe lattice. Parameters: U=2, U=2.2, U=2.4, U=2.6, U=2.8, U=3 b=64, band width W=2 CTQMC scheme with b=64

  13. Complexity of the algorithm At each step, determinant of kxk matrix should be calculated. Fast-update trick allows to do it in k2 operations. Without the sign problem, average perturbation order can be estimated as <k> = - b <W> ~ bN U For Hubbard-like models operation count is the same as in Hirsch-Fye scheme. It is possible to do the calculations for non-local interaction, with the similar operation count, contrary to discrete-time scheme. Complexity of the algorithm is similar one of the discrete scheme, but the advantage is that it naturally describes the interaction nonlocal in time, space, or spin indices. • Other advantages are: • accurate high-frequency tail; • better practical stability of DMFT loops against stochastic noise

  14. Dynamical screening An action with retarded interaction is Split of the interaction into 5 parts allows to get rid of sign problem and implement quantum Wang-Landau sampling The resulting data for G(t) at b=32 U0=0.5 U (upper curve) and U0=0.1 U (lower curve). The analytical predictions are shown with dot lines.

  15. Correlated trimer with rotationally-invariant exchange We consider three atoms with rotationally-invariant exchange in a bath DOS for equilateral triangle (ET) and isosceles triangle (IT) geometries. Parameters: J23=J,J12=J/3, J13=J/3. There are two dependencies in case of IT: one for adatom 1 and another for equivalent adatoms $2$ and $3$. Imaginary part of the Green functions at Matsubara frequencies for the correlated adatom equilateral triangle in the metallic bath. Parameters: U=2, t=0, b=16, m=U/2, J=0.2 (upper panel) and -0.2 (lower panel). The insets show DOS.

  16. Multiorbital CT-QMC: general U-vertex G() Uijkl Udiag  E. Gorelov, et. al. to be published

  17. Existing code Input text file with system-specific parameters Input text file with general parameters KERNEL MODUL (universal) Sampling procedures and auxiliary things USER MODUL (system-specific) H0 and W Self-energy S(iw) Green function G(iw) http://www.ctqmc.ru

  18. Co in Cu 5d-orbitals QMC calculation CT-QMC LDA G(t)  DOS for Co atom in Cu U=4, b = 10 (T ~ 1/40 W) E. Gorelov et al, to be published

  19. Beyond DMFT: diagrams for non-local dual self-energy Lines denote the renormalized Green’s function. Vertexes are vertex parts of DMFT-type impurity problem: are calculated by ct-qmc

  20. Many thanks to A.I. Lichtenstein (Hamburg University)

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